Some remarks on the asymptotic behaviour of the cosmic ray cascade for large depth of the absorber

Conclusions

We have proved (4) and (5).Γ(ε′, 0,x).Γ(ε, 0,x) denotes the conditional probability that we find at least one nucleon of energy >ε′, provided that there is at least one nucleon of energy >ε whereε′ >ε at depthx

Let us now consider the physical meaning of these two theorems.

The relation (4) states: for very large depth of the absorber if we find at least one nucleon of energy >ε we may be sure that the energy of the nucleons lies in the infinitesimal energy range about,ε i.e. (ε, ε - dε).

The relation (5) or (21) states: for very large depth of the absorber if we tind at least one nucleon of energy >ε we may be sure that there is only one nucleon.

Both relations taken together state: if we find for very large depth of the absorber at least one nucleon of energy >ε we may be sure that there is only one nucleon his energy lying in the infinitesimal energy range (ε, ε -dε).

Formulae (5) or (21) give asymptotic solutions for the distribution function. This result may be interesting from the point of view of the theoretical physicist. The experimental physicist, however, requires among other numerical data for large but finite depth of the absorber. Our solution realizes the first step in an approximation procedure for large but finitex. Jt seems very probable that the next step in the approximation will be given by the second factorial moment, the next following by the third and so on. The second part of this paper is devoted to this problem.